Principles of Optics for Engineers Uniting historically different approaches by presenting optical analyses as solutions of Maxwell s equations, this unique book enables students and practicing engineers to fully understand the similarities and differences between the various methods. The book begins with a thorough discussion of plane wave analysis, which provides a clear understanding of optics without considering boundary condition or device configuration. It then goes on to cover diffraction analysis, including a rigorous analysis of TEM waves using Maxwell s equations, and the use of Gaussian beams to analyze different applications. Modes of simple waveguides and fibers are also covered, as well as several approximation methods including the perturbation technique, the coupled mode analysis, and the super mode analysis. Analysis and characterization of guided wave devices, such as power dividers, modulators, and switches, are presented via these approximation methods. With theory linked to practical examples throughout, it provides a clear understanding of the interplay between plane wave, diffraction, and modal analysis, and how the different techniques can be applied to various areas such as imaging, spectral analysis, signal processing, and optoelectronic devices. William S. C. Chang is an Emeritus Professor of the Department of Electrical and Computer Engineering, University of California, San Diego (UCSD). After receiving his Ph.D. from Brown University in 1957, he pioneered maser and laser research at Stanford University, and he has been involved in guided-wave teaching and research at Washington University and UCSD since 1971. He has published over 200 technical papers and several books, including Fundamentals of Guided-Wave Optoelectronic Devices (Cambridge, 2009), Principles of Lasers and Optics (Cambridge, 2005) and RF Photonic Technology in Optical Fiber Links (Cambridge, 2002).
Principles of Optics for Engineers Diffraction and Modal Analysis BY WILLIAM S. C. CHANG University of California, San Diego
University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. Information on this title: /9781107074903 Cambridge University Press 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Chang, William S. C. (William Shen-chie), 1931 Principles of optics for engineers: diffraction and modal analysis / by William S. C. Chang, University of California, San Diego. pages cm ISBN 978-1-107-07490-3 1. Optical engineering. 2. Diffraction. 3. Modal analysis. I. Title. TA1520.C47 2015 535 dc23 2014042202 ISBN 978-1-107-07490-3 Hardback Additional resources for this publication at /9781107074903 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Introduction 1 1 Optical plane waves in an unbounded medium 4 1.1 Introduction to optical plane waves 4 1.1.1 Plane waves and Maxwell s equations 4 (a) The y-polarized plane wave 5 (b) The x-polarized plane wave 6 1.1.2 Plane waves in an arbitrary direction 7 1.1.3 Evanescent plane waves 9 1.1.4 Intensity and power 9 1.1.5 Superposition and plane wave modes 10 (a) Plane waves with circular polarization 10 (b) Interference of coherent plane waves 10 (c) Representation by summation of plane waves 11 1.1.6 Representation of plane waves as optical rays 13 1.2 Mirror reflection of plane waves 14 1.2.1 Plane waves polarized perpendicular to the plane of incidence 14 1.2.2 Plane waves polarized in the plane of incidence 15 1.2.3 Plane waves with arbitrary polarization 15 1.2.4 The intensity 15 1.2.5 Ray representation of reflection 15 1.2.6 Reflection from a spherical mirror 16 1.3 Refraction of plane waves 17 1.3.1 Plane waves polarized perpendicular to the plane of incidence 17 1.3.2 Plane waves polarized in the plane of incidence 19 1.3.3 Properties of refracted and transmitted waves 20 (a) Transmission and reflection at different incident angles 20 (b) Total internal reflection 21 (c) Refraction and reflection of arbitrary polarized waves 21 (d) Ray representation of refraction 21 1.3.4 Refraction and dispersion in prisms 22 (a) Plane wave analysis of prisms 22
vi (b) Ray analysis of prisms 24 (c) Thin prism represented as a transparent layer with a varying index 24 1.3.5 Refraction in a lens 25 (a) Ray analysis of a thin lens 25 (b) Thin lens represented as a transparency with varying index 27 1.4 Geometrical relations in image formation 28 1.5 Reflection and transmission at a grating 30 1.6 Pulse propagation of plane waves 31 Chapter summary 32 2 Superposition of plane waves and applications 34 2.1 Reflection and anti-reflection coatings 34 2.2 Fabry Perot resonance 37 2.2.1 Multiple reflections and Fabry Perot resonance 37 2.2.2 Properties of Fabry Perot resonance 39 2.2.3 Applications of the Fabry Perot resonance 41 (a) The Fabry Perot scanning interferometer 41 (b) Measurement of refractive properties of materials 42 (c) Resonators for filtering and time delay of signals 43 2.3 Reconstruction of propagating waves 43 2.4 Planar waveguide modes viewed as internal reflected plane waves 46 2.4.1 Plane waves incident from the cladding 46 2.4.2 Plane waves incident from the substrate 48 (a) Incident plane waves with sin 1 ðn c =n s Þ < θ s < π=2 48 (b) Incident plane waves with 0 < θ s < sin 1 ðn c =n s Þ 48 2.4.3 Plane waves incident within the waveguide: the planar waveguide modes 48 2.4.4 The hollow dielectric waveguide mode 50 Chapter summary 51 3 Scalar wave equation and diffraction of optical radiation 53 3.1 The scalar wave equation 54 3.2 The solution of the scalar wave equation: Kirchhoff s diffraction integral 55 3.2.1 Kirchhoff s integral and the unit impulse response 57 3.2.2 Fresnel and Fraunhofer diffractions 57 3.2.3 Applications of diffraction integrals 58 (a) Far field diffraction pattern of an aperture 58 (b) Far field radiation intensity pattern of a lens 60
vii (c) Fraunhofer diffraction in the focal plane of a lens 62 (d) The lens viewed as a transformation element 65 3.2.4 Convolution theory and other mathematical techniques 65 (a) The convolution relation 66 (b) Double slit diffraction 66 (c) Diffraction by an opaque disk 67 (d) The Fresnel lens 67 (e) Spatial filtering 67 Chapter summary 71 4 Optical resonators and Gaussian beams 73 4.1 Integral equations for laser cavities 74 4.2 Modes in confocal cavities 75 4.2.1 The simplified integral equation for confocal cavities 75 4.2.2 Analytical solutions of the modes in confocal cavities 77 4.2.3 Properties of resonant modes in confocal cavities 78 (a) The transverse field pattern 78 (b) The resonance frequency 79 (c) The orthogonality of the modes 79 (d) A simplified analytical expression of the field 80 (e) The spot size 81 (f) The diffraction loss 81 (g) The line width of resonances 82 4.2.4 Radiation fields inside and outside the cavity 83 (a) The far field pattern of the TEM modes 84 (b) A general expression for the TEM lm Gaussian modes 84 (c) An example to illustrate confocal cavity modes 85 4.3 Modes of non-confocal cavities 86 4.3.1 Formation of a new cavity for known modes of confocal resonator 86 4.3.2 Finding the virtual equivalent confocal resonator for a given set of reflectors 88 4.3.3 A formal procedure to find the resonant modes in non-confocal cavities 89 4.3.4 An example of resonant modes in a non-confocal cavity 91 4.4 The propagation and transformation of Gaussian beams (the ABCD matrix) 91 4.4.1 A Gaussian mode as a solution of Maxwell s equation 92 4.4.2 The physical meaning of the terms in the Gaussian beam expression 94 4.4.3 The analysis of Gaussian beam propagation by matrix transformation 95 4.4.4 Gaussian beam passing through a lens 97
viii 4.4.5 Gaussian beam passing through a spatial filter 98 4.4.6 Gaussian beam passing through a prism 100 4.4.7 Diffraction of a Gaussian beam by a grating 102 4.4.8 Focusing a Gaussian beam 103 4.4.9 An example of Gaussian mode matching 104 4.4.10 Modes in complex cavities 105 4.4.11 An example of the resonance mode in a ring cavity 106 Chapter summary 107 5 Optical waveguides and fibers 109 5.1 Introduction to optical waveguides and fibers 109 5.2 Electromagnetic analysis of modes in planar optical waveguides 112 5.2.1 The asymmetric planar waveguide 112 5.2.2 Equations for TE and TM modes 112 5.3 TE modes of planar waveguides 113 5.3.1 TE planar guided-wave modes 114 5.3.2 TE planar guided-wave modes in a symmetrical waveguide 115 5.3.3 The cut-off condition of TE planar guided-wave modes 117 5.3.4 An example of TE planar guided-wave modes 118 5.3.5 TE planar substrate modes 119 5.3.6 TE planar air modes 119 5.4 TM modes of planar waveguides 121 5.4.1 TM planar guided-wave modes 121 5.4.2 TM planar guided-wave modes in a symmetrical waveguide 122 5.4.3 The cut-off condition of TM planar guided-wave modes 123 5.4.4 An example of TM planar guided-wave modes 123 5.4.5 TM planar substrate modes 124 5.4.6 TM planar air modes 125 5.4.7 Two practical considerations for TM modes 126 5.5 Guided waves in planar waveguides 126 5.5.1 The orthogonality of modes 126 5.5.2 Guided waves propagating in the y z plane 127 5.5.3 Convergent and divergent guided waves 127 5.5.4 Refraction of a planar guided wave 128 5.5.5 Focusing and collimation of planar guided waves 129 (a) The Luneberg lens 129 (b) The geodesic lens 129 (c) The Fresnel diffraction lens 130 5.5.6 Grating diffraction of planar guided waves 131 5.5.7 Excitation of planar guided-wave modes 134 5.5.8 Multi-layer planar waveguides 135
ix 5.6 Channel waveguides 135 5.6.1 The effective index analysis 136 5.6.2 An example of the effective index method 140 5.6.3 Channel waveguide modes of complex structures 141 5.7 Guided-wave modes in optical fibers 142 5.7.1 Guided-wave solutions of Maxwell s equations 142 5.7.2 Properties of the modes in fibers 144 5.7.3 Properties of optical fibers in applications 145 5.7.4 The cladding modes 146 Chapter summary 146 6 Guided-wave interactions 148 6.1 Review of properties of the modes in a waveguide 149 6.2 Perturbation analysis 150 6.2.1 Derivation of perturbation analysis 150 6.2.2 A simple application of perturbation analysis: perturbation by a nearby dielectric 152 6.3 Coupled mode analysis 153 6.3.1 Modes of two uncoupled parallel waveguides 153 6.3.2 Modes of two coupled waveguides 154 6.3.3 An example of coupled mode analysis: the grating reflection filter 155 6.3.4 Another example of coupled mode analysis: the directional coupler 160 6.4 Super mode analysis 163 6.5 Super modes of two parallel waveguides 163 6.5.1 Super modes of two well-separated waveguides 164 6.5.2 Super modes of two coupled waveguides 164 6.5.3 Super modes of two coupled identical waveguides 166 (a) Super modes obtained from the effective index method 166 (b) Super modes obtained from coupled mode analysis 168 6.6 Directional coupling of two identical waveguides viewed as super modes 169 6.7 Super mode analysis of the adiabatic Y-branch and Mach-Zehnder interferometer 170 6.7.1 The adiabatic horn 170 6.7.2 Super mode analysis of a symmetric Y-branch 171 (a) A single-mode Y-branch 171 (b) A double-mode Y-branch 173 6.7.3 Super mode analysis of the Mach Zehnder interferometer 173 Chapter summary 175
x 7 Passive waveguide devices 176 7.1 Waveguide and fiber tapers 176 7.2 Power dividers 176 7.2.1 The Y-branch equal-power splitter 177 7.2.2 The directional coupler 177 7.2.3 The multi-mode interference coupler 178 7.2.4 The Star coupler 182 7.3 The phased array channel waveguide frequency demultiplexer 186 7.4 Wavelength filters and resonators 188 7.4.1 Grating filters 188 7.4.2 DBR resonators 189 7.4.3 The ring resonator wavelength filter 189 (a) Variable-gap directional coupling 190 (b) The resonance condition of the couple ring 191 (c) Power transfer 192 (d) The free spectral range and the Q-factor 192 7.4.4 The ring resonator delay line 194 Chapter summary 195 8 Active opto-electronic guided-wave components 196 8.1 The effect of electro-optical χ 197 8.1.1 Electro-optic effects in plane waves 197 8.1.2 Electro-optic effects in waveguides at low frequencies 198 (a) Effect of Δχʹ 198 (b) Effect of Δχʹʹ 199 8.2 The physical mechanisms to create Δχ 200 8.2.1 Δχʹ 200 (a) The LiNbO 3 waveguide 202 (b) The polymer waveguide 203 (c) The III V compound semiconductor waveguide 203 8.2.2 Δχʹʹ in semiconductors 205 (a) Stimulated absorption and the bandgap 205 (b) The quantum-confined Stark effect, QCSE 206 8.3 Active opto-electronic devices 211 8.3.1 The phase modulator 211 8.3.2 The Mach Zhender modulator 212 8.3.3 The directional coupler modulator/switch 213 8.3.4 The electro-absorption modulator 214 8.4 The traveling wave modulator 215 Chapter summary 217 Appendix 219 Index 225